#6937517

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1291 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,29,34,30,33,42,47)(27,31,37,45,36,44,52,55)(28,38,39,46,40,48,41,32)(35,49,51,54,50,56,53,43);(1,2)(3,6)(4,5)(7,8)(10,14)(12,19)(15,20)(18,23)(25,27)(26,31)(28,35)(29,37)(30,36)(32,43)(33,44)(34,45)(38,49)(39,51)(40,50)(41,53)(42,52)(46,54)(47,55)(48,56);(2,5)(3,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,40)(26,32,33,46)(27,35,36,50)(29,41,42,39)(31,43,44,54)(34,48,47,38)(37,53,52,51)(45,56,55,49);(1,3,4,7)(2,6,5,8)(10,14)(11,17)(16,21)(18,23)(27,36)(31,44)(35,50)(37,52)(43,54)(45,55)(49,56)(51,53);(2,5)(6,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(26,33)(31,44)(32,46)(34,47)(38,48)(43,54)(45,55)(49,56)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844713);